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Impulse and time integration

Impulse measures the accumulated effect of force over a time interval. A large force acting briefly and a smaller force acting longer can produce the same change in momentum. This is why collision analysis can often use the area under a force-time curve without resolving every detail of material deformation.

Impulse-momentum theorem

Newton's second law in momentum form is

\[\mathbf F_{\mathrm{net}}=\frac{d\mathbf p}{dt}\]

Integrating from initial time \(t_i\) to final time \(t_f\) gives

\[\mathbf J=\int_{t_i}^{t_f}\mathbf F_{\mathrm{net}}(t)\,dt=\mathbf p_f-\mathbf p_i=\Delta\mathbf p\]

Impulse \(\mathbf J\) has SI units of newton second. Since \(1\,\mathrm N=1\,\mathrm{kg\,m/s^2}\), a newton second equals a kilogram metre per second, the unit of momentum.

Impulse is a vector. In several dimensions, integrate each force component with its sign. The integral of force magnitude is generally not the magnitude of impulse because the force direction may change during the interval.

Geometric meaning

On a graph of one force component against time, signed impulse is the signed area between the curve and the time axis. Areas above the axis contribute positively; areas below contribute negatively under the chosen direction convention. A tall narrow pulse and a shorter wide pulse can have equal areas.

For a constant net force,

\[\mathbf J=\mathbf F_{\mathrm{net}}(t_f-t_i)=\mathbf F_{\mathrm{net}}\Delta t\]

For any force history, define average force over the interval by

\[\mathbf F_{\mathrm{avg}}=\frac{1}{\Delta t}\int_{t_i}^{t_f}\mathbf F(t)\,dt=\frac{\mathbf J}{\Delta t}\]

Average force is not generally the arithmetic average of the largest and smallest force. That shortcut works only for special curve shapes, such as a linear ramp over the full interval.

A stopping example

A ball of mass

\(0.20\,\mathrm{kg}\)

moves right at \(15\,\mathrm{m/s}\) and is brought to rest. Taking right as positive,

\[J=\Delta p=m(v_f-v_i)=(0.20)(0-15)=-3.0\ \mathrm{N\,s}\]

The negative sign means the net impulse points left. If contact lasts \(0.050\,\mathrm s\), the average net force is

\[F_{\mathrm{avg}}=\frac{-3.0}{0.050}=-60\ \mathrm N\]

Doubling the stopping time while keeping the same initial and final momenta halves the required average force. This explains the protective value of padding, but the peak force still depends on the shape of the force-time curve.

Integrating sampled force data

Measured or calculated forces are often available only at sample times. If a force value represents the left endpoint of each interval, the left-rectangle estimate is

\[\mathbf J\approx\sum_{n=0}^{N-1}\mathbf F_n\Delta t_n\]

If values represent endpoints of a smooth curve, the trapezoidal estimate is

\[\mathbf J\approx\sum_{n=0}^{N-1}\frac{\mathbf F_n+\mathbf F_{n+1}}{2}\Delta t_n\]

Do not apply a method without knowing what each sample means. A sensor may report an instantaneous sample, an average over an acquisition window, or a filtered value. If it already reports interval averages, multiplying each average by its interval duration is the natural accumulation.

import numpy as np

times_s = np.linspace(0, 0.2, 101)
forces_n = 100 * np.sin(np.pi * times_s / 0.2)
impulse_ns = np.trapezoid(forces_n, times_s)
print(f"Impulse: {impulse_ns:.6f} N s")

Uneven sample spacing requires each \(\Delta t_n\) to be used. Multiplying every force by one nominal interval can bias the result when timestamps jitter or samples are missing.

Accuracy and convergence

For a smooth force, reducing sample spacing should make reasonable quadrature methods approach the same impulse. Compare results at fixed physical endpoints. Midpoint and trapezoidal rules are typically more accurate than an endpoint rectangle at the same uniform spacing, but sharp discontinuities and unresolved pulses can dominate formal order.

A collision pulse narrower than the sampling interval may be missed completely. No quadrature formula can reconstruct information that was never sampled without additional assumptions. Increase acquisition rate, measure momentum change independently, or use a calibrated impulse sensor.

exact_impulse = 40 / np.pi
for samples in [21, 41, 81, 161]:
    times = np.linspace(0, 0.2, samples)
    force = 100 * np.sin(np.pi * times / 0.2)
    estimate = np.trapezoid(force, times)
    print(samples, f"impulse error: {abs(estimate-exact_impulse):.6f} N s")

Keep endpoint choice fixed. Expanding or shrinking the contact interval while refining sampling changes both numerical resolution and the physical integral.

Internal and external impulses

Two colliding bodies exert equal-and-opposite internal impulses:

\[\mathbf J_{12}=-\mathbf J_{21}\]

Their sum cancels in the momentum balance for the two-body system. For either body alone, the contact impulse is external to that smaller system and changes its momentum.

A wall, track, fluid, or field can provide additional impulse. The correct system balance is

\[\Delta\mathbf P_{\mathrm{system}}=\mathbf J_{\mathrm{external}}\]

If gravity acts during a short horizontal collision, its vertical impulse might matter even when horizontal momentum is approximately conserved. Evaluate vector components instead of calling the event simply isolated or non-isolated.

Instantaneous impulses as an idealization

An ideal impulse can be modeled as changing momentum at one instant while position remains continuous. This is a useful approximation when contact duration is much shorter than every other timescale of interest. It predicts before-and-after velocity but not peak force, stress, deformation, sound, or detailed contact duration.

A real contact force is finite and time-dependent. Resolving it requires material stiffness, damping, geometry, and sufficiently small time intervals. Different force histories can have the same impulse and therefore the same final momentum while producing very different peak loads and damage.

Momentum updates

A first-order discrete momentum update is

\[\mathbf p_{n+1}=\mathbf p_n+\mathbf F_n\Delta t\]

This has the same form as a left-rectangle impulse estimate. It is exact for constant force over the interval but approximate when force changes. Higher-order time integrators combine several force evaluations; the effective impulse must match the method's stages.

For interacting bodies, calculate pair force once and apply opposite momentum increments from that same value. Updating one body before evaluating the other mixes times and can create false total impulse.

Uncertainty in measured impulse

Force calibration, baseline offset, timestamp precision, filtering, and sensor bandwidth all affect an experimental integral. A small constant force offset accumulates with duration and can dominate a long interval. Estimate the pre-contact baseline and report whether it was subtracted.

Compare force-integrated impulse with independently measured \(\Delta\mathbf p\). Agreement within combined uncertainty supports both measurements. Disagreement may reveal missed high-frequency force, incorrect mass, velocity-estimation error, or an omitted external force.

Reliability checks

Impulse connects a detailed force history to a compact change in state. Its integral is simple, but a trustworthy value depends on vector signs, system boundaries, sample meaning, and enough time resolution to capture the force history.

A worked Python notebook

Integrate a half-sine contact force lasting 0.2 seconds. The question for this calculation is precise: how does force change when we change time during force pulse, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The area is impulse, not energy; the pulse is defined only over its contact interval. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 0.2 s. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for time during force pulse, expressed in s. The output is force, expressed in N. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 100 \sin{\left(5 \pi q \right)}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 0.1 s, the expression gives 100 N. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("100*sin(pi*q/0.2)", locals={"q": q})
reference_input = 0.1
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted force:", reference_output, "N")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0 N. The midpoint, at 0.1 s, gives 100 N. The upper endpoint gives 0 N. Read these together with the assumptions: they describe different choices of time during force pulse within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 0.1, 0.2])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} s -> {output_value:.6g} N")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows time during force pulse; the vertical axis shows force. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 0.2, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Time during force pulse (s)")
ax.set_ylabel("Force (N)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does force change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.1 s is 0 in those ratio units. Locally, the output has no first-order change as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 500 \pi \cos{\left(5 \pi q \right)}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.101
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "N")
print("Full expression:", full_prediction, "N")
print("Approximation difference:", full_prediction - linear_prediction, "N")

Here the changed input is 0.101 s. The tangent estimate is 100 N, while the full expression gives 99.9877 N. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 0.2
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of time during force pulse inside the stated interval, predict whether force will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The area is impulse, not energy; the pulse is defined only over its contact interval. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of N: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(0.2) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 0.2, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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